{"id":729,"date":"2020-12-07T16:35:39","date_gmt":"2020-12-07T16:35:39","guid":{"rendered":"https:\/\/pickedshares.com\/?p=729"},"modified":"2021-05-08T11:52:47","modified_gmt":"2021-05-08T11:52:47","slug":"engineering-mechanics-1-exercise-19-truss-member-forces-and-bearing-loads","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-19-truss-member-forces-and-bearing-loads\/","title":{"rendered":"Truss, member forces and bearing loads"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">Task<\/h3>\n\n\n\n<p>The force F is acting under an angle on a truss. The truss has a locating bearing and a floating bearing. Calculate the bearing loads and the member forces for the truss.<\/p>\n\n\n\nF = 10 kN<br>\n\u03b1 = 60\u00b0<br>\na = 2 m\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"775\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-1024x775.jpg\" alt=\"Truss with fixed bearing and floating bearing under external load\" class=\"wp-image-435\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-1024x775.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-300x227.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-768x581.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-400x303.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19-800x605.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19.jpg 1146w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Truss with fixed bearing and floating bearing under external load<\/figcaption><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p>The following video is in german language.<\/p>\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"lyte-wrapper\" title=\"Technische Mechanik 1, &Uuml;bung 19 - Stabwerk\" style=\"width:640px;max-width:100%;margin:5px auto;\"><div class=\"lyMe hidef\" id=\"WYL_jktQHs2yWKI\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><div><meta itemprop=\"thumbnailUrl\" content=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FjktQHs2yWKI%2Fmaxresdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/jktQHs2yWKI\" \/><meta itemprop=\"duration\" content=\"PT21M4S\" \/><meta itemprop=\"uploadDate\" content=\"2020-11-26T14:27:17Z\" \/><\/div><div id=\"lyte_jktQHs2yWKI\" data-src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FjktQHs2yWKI%2Fmaxresdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">Technische Mechanik 1, \u00dcbung 19 - Stabwerk<\/div><\/div><div class=\"play\"><\/div><div class=\"ctrl\"><div class=\"Lctrl\"><\/div><div class=\"Rctrl\"><\/div><\/div><\/div><noscript><a href=\"https:\/\/youtu.be\/jktQHs2yWKI\" rel=\"nofollow\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FjktQHs2yWKI%2F0.jpg\" alt=\"Technische Mechanik 1, &Uuml;bung 19 - Stabwerk\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"Berechnung der Lagerreaktionen und Stabkr\u00e4fte eines Stabwerks, an dem eine Kraft unter einem gegebenen Winkel angreift. In diesem Fall wird das Knotenpunktverfahren angewendet. Die Gleichungen bzw. Stabkr\u00e4fte werden nach Aufstellung der Gleichgewichtsbedingungen &quot;manuell&quot; gel\u00f6st, also ohne Gau\u00df-Algorithmus. Diese \u00dcbung ist auch unter pickedshares.com zu finden.\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px auto;\"><\/div><figcaption>Solution video for exercise 19 - truss<\/figcaption><\/figure>\n\n\n<h4 class=\"wp-block-heading\">Sketches<\/h4>\n\n\n\n<p>The following sketches show the <a href=\"https:\/\/pickedshares.com\/en\/reaction-forces-in-bearings-joints-and-guidances\/\">bearing reaction forces<\/a>, the member and node numbering and the location and direction of the forces at each node.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"817\" height=\"872\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved.jpg\" alt=\"Truss with single nodes and forces\" class=\"wp-image-444\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved.jpg 817w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved-281x300.jpg 281w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved-768x820.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved-400x427.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-19_solved-800x854.jpg 800w\" sizes=\"auto, (max-width: 817px) 100vw, 817px\" \/><figcaption>Truss with single nodes and forces<\/figcaption><\/figure>\n\n\n\n<p>The forces are solved symbolic first, then the real numbers are set in to calculate the values. Since the angles of the bars are always 45 \u00b0, the <a href=\"https:\/\/pickedshares.com\/en\/important-angles-and-related-results-of-the-trigonometric-functions\/\">alternative notation of the trigonometric functions<\/a> is used.<\/p>\n\n\n\n<!DOCTYPE html>\n<!DOCTYPE html>\n<html lang=\"\">\n  <head>\n    <meta http-equiv=\"Content-Type\" content=\"text\/html; charset=utf-8\"\/>\n    <script type=\"text\/x-mathjax-config\">  MathJax.Hub.Config({\n    displayAlign: \"left\",\n    context: \"MathJax\",\n    TeX: {TagSide: \"left\"}\n  })\n<\/script>\n    <script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n  <\/head>\n  <body>\n    <!-- ****************************************************** -->\n    <!-- *        Created with wxMaxima version 20.06.6       * -->\n    <!-- ****************************************************** -->\n    <noscript>\n      <div class=\"error message\">\n        <p>Please enable JavaScript in order to get a 2d display of the equations embedded in this web page.<\/p>\n      <\/div>\n    <\/noscript>\n    <p hidden=\"hidden\">\\(      \\DeclareMathOperator{\\abs}{abs}\n      \\newcommand{\\ensuremath}[1]{\\mbox{$#1$}}\n\\)<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\"><h4>Bearing reactions<\/h4>\n<\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">Sum of all forces in y-direction\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{1} 0=-F \\sin{\\left( \\alpha \\right) }+{F_{\\mathit{By}}}+{F_{\\mathit{Ay}}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Sum of all forces in x-direction\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{2} 0={F_{\\mathit{Bx}}}-F \\cos{\\left( \\alpha \\right) }\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Sum of all moments around point A\n<\/div>\n\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{3} 0=-2 F a \\sin{\\left( \\alpha \\right) }+F a \\cos{\\left( \\alpha \\right) }+3 {F_{\\mathit{By}}} a\\]\n<\/p>\n<\/div>\n    <!-- Text cell -->\n<div class=\"comment\"><h4>Node equations<\/h4>\n<\/div>\n    <div class=\"comment\">Node I x\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{4} 0=\\frac{{S_4}}{\\sqrt{2}}+{S_1}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node I y\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{5} 0=\\frac{{S_4}}{\\sqrt{2}}+{F_{\\mathit{Ay}}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node II x\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{6} 0=\\frac{{S_6}}{\\sqrt{2}}+{S_2}-{S_1}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node II y\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{7} 0=\\frac{\\mathit{S6}}{\\sqrt{2}}+\\mathit{S5}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node III x\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{8} 0={S_3}-{S_2}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node III y\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{9} 0={S_7}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node IV x\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{10} 0=-\\frac{{S_8}}{\\sqrt{2}}-{S_3}+{F_{\\mathit{Bx}}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node IV y\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{11} 0=\\frac{{S_8}}{\\sqrt{2}}+{F_{\\mathit{By}}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node V x\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{12} 0={S_9}-\\frac{{S_4}}{\\sqrt{2}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node V y\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{13} 0=-{S_5}-\\frac{{S_4}}{\\sqrt{2}}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Node VI x\n<\/div>\n\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{14} 0=-F \\cos{\\left( \\alpha \\right) }-{S_9}+\\frac{{S_8}}{\\sqrt{2}}-\\frac{{S_6}}{\\sqrt{2}}\\]\n<\/p>\n<\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">Node VI y\n<\/div>\n\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{15} 0=-F \\sin{\\left( \\alpha \\right) }-\\frac{{S_8}}{\\sqrt{2}}-{S_7}-\\frac{{S_6}}{\\sqrt{2}}\\]\n<\/p>\n<\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">Equation (2) solved for F<sub>Bx<\/sub>\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{16} {F_{\\mathit{Bx}}}=F \\cos{\\left( \\alpha \\right) }\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Equation (9) gives Zero-member S<sub>7<\/sub>\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{17} {S_7}=0\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">Equation (3) solved for F<sub>By<\/sub>\n<\/div>\n\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{18} {F_{\\mathit{By}}}=\\frac{2 F \\sin{\\left( \\alpha \\right) }-F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n<\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">F<sub>By<\/sub> set in equation (1) ...\n<\/div>\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{19} 0=\\frac{2 F \\sin{\\left( \\alpha \\right) }-F \\cos{\\left( \\alpha \\right) }}{3}-F \\sin{\\left( \\alpha \\right) }+{F_{\\mathit{Ay}}}\\]\n<\/p>\n<\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">... and solved for F<sub>Ay<\/sub>\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{20} {F_{\\mathit{Ay}}}=\\frac{F \\sin{\\left( \\alpha \\right) }+F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Text cell -->\n    <div class=\"comment\">In the following, the forces solved so far are inserted into the remaining equations and the unknowns are solved\n<\/div>\n\n<div style=\"overflow:auto;\">\n    <!-- Code cell -->\n    <p>\\[\\tag{21} {S_4}=-\\frac{\\sqrt{2} F \\sin{\\left( \\alpha \\right) }+\\sqrt{2} F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{22} {S_1}=\\frac{F \\sin{\\left( \\alpha \\right) }+F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{23} {S_9}=-\\frac{F \\sin{\\left( \\alpha \\right) }+F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{24} {S_5}=\\frac{F \\sin{\\left( \\alpha \\right) }+F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{25} {S_8}=-\\frac{{{2}^{\\frac{3}{2}}} F \\sin{\\left( \\alpha \\right) }-\\sqrt{2} F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{26} {S_3}=\\frac{2 F \\sin{\\left( \\alpha \\right) }+2 F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{27} {S_6}=-\\frac{\\sqrt{2} F \\sin{\\left( \\alpha \\right) }+\\sqrt{2} F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{28} {S_2}=\\frac{2 F \\sin{\\left( \\alpha \\right) }+2 F \\cos{\\left( \\alpha \\right) }}{3}\\]\n<\/p>\n    <\/div>\n    <!-- Text cell -->\n    <div class=\"comment\">Resulting member forces and bearing loads\n<\/div>\n    <!-- Code cell -->\n    <p>\\[\\tag{29} {S_1}=4.55 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{30} {S_2}=9.11 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{31} {S_3}=9.11 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{32} {S_4}=-6.44 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{33} {S_5}=4.55 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{34} {S_6}=-6.44 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{35} {S_7}=0.0\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{36} {S_8}=-5.81 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{37} {S_9}=-4.55 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{38} {F_{\\mathit{Ay}}}=4.55 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{39} {F_{\\mathit{By}}}=4.11 \\mathit{kN}\\]\n<\/p>\n    <!-- Code cell -->\n    <p>\\[\\tag{40} {F_{\\mathit{Bx}}}=5.0 \\mathit{kN}\\]\n<\/p>\n    <hr\/>\n    \n  <\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Truss, member forces and bearing loads\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-19-truss-member-forces-and-bearing-loads\/\" aria-label=\"Read more about Truss, member forces and bearing loads\">Read 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